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It CAN also be high but the two are not related. As DrWhoF stated, gain is a measure of by how much the input signal is enlarged, swing is the limit of how big it can be at the output point.
For example, suppose you are using a 9V supply and your input signal is 100mV. A gain of 10 might give you 1V output but the same amplifier with 1V input wouldn't be able to give you 10V output because it electrically can't produce (swing it's output) by 10V from a 9V supply.
The amount of swing can not exceed the supply and the actual amount will depend upon the design of the amplifier stage itself. In reality, the swing rarely reaches 100% of supply, especially in linear amplifiers. A more typical figure is probably 75% but often the design objective isn't always to maximize the output anyway.
Gain is the measure of amplification of the input signal.
Swing refers to how high Vout can go. Sometimes you may hear the term "rail-to-rail" swing, which means that Vout can go as high as the positive supply voltage and as low as the negative supply voltage (or ground if negative supply isn't used).
Well, an amplifier amplifies your signal ,it means the magnitude and power of the output signal is usually greater than the input signal (Output signal is stronger). For a voltage amplifier we can say Gain=Vout/Vin ,which shows how much the amplifier can make the signal stronger. But it's very important to know , the amplifier is able to amplify the signal only AS LONG AS the amplifier (the BJT) doesn't enter SATURATION area. And VOLTAGE SWING relates to this. Voltage Swing tells you the possible range of the ouput voltage of the amplifier. It means that the output voltage (amplitude of a sine wave,for instance) can NOT be greater or smaller than certain values that are called Swing voltages. When the input voltage (amplitude of input) is too high, the transisitor(s) will enter saturation area and the formula Gain=Vout/Vin won't be valid anymore (there won't be more amplification) so the output voltage never increases more than swing voltages.